# Confused by these proton decay diagrams

1. Jul 30, 2015

### Anchovy

I have some confusion over some diagrams involved in proton decays in SU(5). I've seen a few like this that look weird to me. Here is one I found where the colored Higgs triplet does the job:

Straight away I don't know what to make of this because I've only ever seen diagrams where fermions are connected by one exchange boson. Here there are two, the $T$ and $\overline{T}$.

If I'm not mistaken, $T$ is meant to have a charge of -1/3 and $\overline{T}$ has +1/3. So if I specify the left hand side quarks with two of the three (uud) proton constituents, $u^{+2/3}$ and $d^{-1/3}$, then the charge from those quarks going into the left vertex is +1/3 overall. So there should be a $\overline{T}$ coming out of that vertex since it also has charge +1/3. But then the arrows look wrong since the arrows on the u, d should be pointing to the right?
Anyway, then the right hand side vertex would have $\overline{T}$ going in, so then the leading decay mode in this situation ($p \rightarrow K^{+} \overline{\nu}$) would invove the Q on the right hand side being an $\overline{s}$ with charge of +1/3 (with the spectator u quark (which isn't shown) combining with the $\overline{s}$ to produce the $K^{+}$), and the antilepton L being neutral, ie. $\overline{\nu}$.

So the diagram seems to make sense if I ignore one of the bosons. Has whoever drew this diagram combined two different interactions into the same diagram here, and if so, do the arrows really make sense? Or am I misinterpreting the $T$ / $\overline{T}$ line in this diagram?

Here's another one, this time a SUSY diagram:

The $u^{+2/3}$ and $d^{-1/3}$ put an overall +1/3 charge into the red dot vertex, the anti-stop (anti based on the arrow direction) has a -2/3 charge and the anti-stau a +1 charge, so the red dot vertex has overall +1/3 coming out of it too. So far so good.
Then we get to the vertices with the Higgsino. At the upper vertex, there's a -2/3 charge going in from the stop, and the anti-strange quark has charge +1/3, so the Higgsino must be a -1 charge (ie. it must be the anti-particle of the superpartner to the $H^{+}$ in the $(H^{+}, H^{0})$ SU(2) doublet within the SU(5) $\psi_{\textbf{5}}$ multiplet). Similarly the +1 of the anti-stau and the -1 of the anti-Higgsino cancel to give the electrically neutral antineutrino.

--> Assuming the above interpretation is correct, my issues with this are:
1) Is there a reason why this diagram, meant to indicate proton decay, is set up to be read left-to-right rather than right-to-left?
2) I believe this decay is described by what gets referred to as a 'dimension 5' operator. I think that refers to what is happening at the red dot. However, I don't understand the '5' part. There are only 4 fields involved $(u, d, \tilde{t},\tilde{\tau})$. What is '5-dimensional' about this situation?

2. Jul 30, 2015

### fzero

There's a reason why in the other thread I wanted you to work from first principles. If you are just given diagrams that you don't understand, you're not going to learn as much by having someone throw a few facts at you as you would if you actually worked out where those diagrams came from via the Lagrangian. So for instance, your question about the Higgs diagram is resolved as a simple issue of the theory of a complex scalar field. But since you have not tried to understand the GUT Lagrangian, you do not know the propagator, Feynman rules, etc. So naturally you have questions when you see diagrams written down. But you are not in a good position to absorb the information since you don't have the fundamentals in mind to build on.

I do not want you to take this as harsh criticism and be discouraged. Instead I would like to you be encouraged to step back and fill the gaps in your understanding.

3. Jul 30, 2015

### Anchovy

No no criticism is welcome, I want to understand, I'm just trying to go in ten different directions at once at the moment. I'll get back to the other thread.

4. Jul 30, 2015

### fzero

I will try to briefly answer your questions, but, as a I said, I think you should work out all of the Feynman rules and then come back to them yourself.

The vertex on the LHS comes from an SU(5) Yukawa coupling
$$\epsilon^{ijklm} \psi_{ij} \psi_{kl} \mathcal{H}_m \rightarrow \epsilon^{abc} q_{a1} q_{b2} T_c,$$
where the 2nd term is just those involving SU(3) triplets. I'm using the notation from the other thread, so $q_{a1}$ is up-type and $q_{a2}$ is down. As needed, the U(1) charges are balanced when we draw the arrows on the Feynman diagram either all in or all out. If we want to flip an arrow, we need to compensate by attaching the antiparticle to the end of the leg. Since we want quarks and not antiquarks, it would be best to draw this with the lines pointing in.

The vertex on the RHS comes from
$$\psi^i \psi_{ij} \bar{\mathcal{H}}^j \rightarrow \ell^\alpha q_{a\alpha} \bar{T}^a.$$
If we want an antiquark going out to the right, then we should draw the arrrows headed in. So if we draw just the $T-\bar{T}$ lines, we have

$T$ -----<--------X-------->------$\bar{T}$

This is a correct ordering for the mass insertion, so we conclude that the version of the diagram you included with all arrows reversed is consistent. Evidently whoever wrote that diagram used a backwards convention where an arrow headed out was a particle.

5. Jul 30, 2015

### fzero

We already established for the other diagram that the artist has some backward conventions. At the risk of being culturally insensitive, perhaps his or her native language is read from right to left. The conventions shouldn't be a problem once we've sussed them out.

The mass dimension of a scalar field in 4D is 1, while that of a fermion is 3/2. The mass dimension of the operator represented by the red dot is $1 + 1 + 3/2 + 3/2 = 5$.